Seismic Tomography

Most people know someone who's had a "CAT Scan." That doesn't mean
looking around to see if you let the cat out. CAT stands for Computer Assisted
Tomography. You can send signals through an object in different directions and
add the signals to construct a cross section of the object. If the object is a
human body, doing it with X-rays is a good deal less drastic than doing it with
a scalpel. Other radiation can be used as well; PET scans use positron emission.
In the earth, we can apply the method using seismic waves.

Example 1: Finding Missouri

Imagine you had no way of finding out where Missouri was except by
plotting the paths of satellites and knowing they had either passed over Missouri (red) or
not passed over it (blue).

The concentration of red in the right center suggests the
location of Missouri.

We can achieve a bit more precision by noting that all blue lines pass
outside Missouri and therefore all intersections with a blue line must lie
outside Missouri. These are shown with blue dots.

Only intersections between two red
lines can be within Missouri (red dots). Note that some lie very far from the main
cluster. We suspect these are artifacts because all the other nearby dots are blue. (But
what if the area we were trying to locate really had tiny outlying pieces?)

The main cluster is outlined here in yellow.

This map shows the actual location of Missouri in gray. We could have
gotten a more precise estimate with more closely-spaced lines.

Example 2: Finding Texas

Here's an example that is actually a little closer to the way seismic
tomography is actually done.

Here we've put a grid over the United States. Imagine we
send signals along the grid rows and count the number of squares where we
find Texas.

We go back through the grid and sum the vertical and horizontal signals.
The highest values are the most likely location.

We can also count squares in diagonal directions as well.

Here we sum up the signals in all four directions. We get a sharper fix
on location and also a more precise idea of the shape. A finer grid would
have improved the resolution.

Here's a bit more ambitious example. We'll try to reconstruct a moderately
complex image. Instead of summing numbers in rows, we'll smear the image in
different directions. The brightness represents the strength of the signal. For
example, in the first image, we know there's a large amount of image along the
bright band, but we don't know where. It's somewhere along that band. The other
scans will help identify exactly where.

We can also identify the directions with zero value. We know that no place along
these directions can include any part of the image. This won't work if the
pattern we're trying to reconstruct has positive and negative values, but this
is a simple approximation.

Now we sum up the scans in pairs. First we'll sum 0 and 90 degrees, 30 + 120, 45 +
135, and 60 + 150.

Next we sum those sums. We'll sum 0+45+90+135 and 30+60+120+150

Finally we sum those images to get a final result, and mask off the areas
where we know there is no image.

We can see there's a central mass with a smaller mass to the upper
right, a short projection to the upper left, and projections below with
a vacant space in between. The actual scanned image is at right.

This
is a crude analogy but it manages to capture the main elements of the
shape.